Almost Prime Numbers

A k-Almost Prime Number is a number having exactly k prime factors (not necessary distinct).

For example,

2, 3, 5, 7, 11 ….(in fact all prime numbers) are 1-Almost Prime Numbers as they have only 1 prime factors (which is themselves).



4, 6, 9…. are 2-Almost Prime Numbers as they have exactly 2 prime factors (4 = 2*2, 6 = 2*3, 9 = 3*3)

Similarly 32 is a 5-Almost Prime Number (32 = 2*2*2*2*2) and so is 72 (2*2*2*3*3)

All the 1-Almost Primes are called as Prime Numbers and all the 2-Almost Prime are called as semi-primes.

The task is to print first n numbers that are k prime.

Examples:

Input : k = 2, n = 5
Output : 4 6 9 10 14
4 has two prime factors, 2 x 2
6 has two prime factors, 2 x 3
Similarly, 9(3 x 3), 10(2 x 5) and 14(2 x 7)

Input : k = 10, n = 2
Output : 1024 1536
1024 and 1536 are first two numbers with 10
prime factors.

We iterate natural numbers and keep printing k-primes till the count of printed k-primes is less than or equal to n. To check if a number is k-prime, we find count of prime factors and if the count is k we consider the number as k-prime.
Below is the implementation of above idea :

C

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// Program to print first n numbers that are k-primes
#include<bits/stdc++.h>
  
// A function to count all prime factors of a given number
int countPrimeFactors(int n)
{
    int count = 0;
  
    // Count the number of 2s that divide n
    while (n%2 == 0)
    {
        n = n/2;
        count++;
    }
  
    // n must be odd at this point. So we can skip one
    // element (Note i = i +2)
    for (int i = 3; i <= sqrt(n); i = i+2)
    {
        // While i divides n, count i and divide n
        while (n%i == 0)
        {
            n = n/i;
            count++;
        }
    }
  
    // This condition is to handle the case whien n is a
    // prime number greater than 2
    if (n > 2)
        count++;
  
    return(count);
}
  
// A function to print the first n numbers that are
// k-almost primes.
void printKAlmostPrimes(int k, int n)
{
    for (int i=1, num=2; i<=n; num++)
    {
        // Print this number if it is k-prime
        if (countPrimeFactors(num) == k)
        {
            printf("%d ", num);
  
            // Increment count of k-primes printed
            // so far
            i++;
        }
    }
    return;
}
  
/* Driver program to test above function */
int main()
{
    int n = 10, k = 2;
    printf("First %d %d-almost prime numbers : \n",
           n, k);
    printKAlmostPrimes(k, n);
    return 0;
}

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Java

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// Program to print first n numbers that 
// are k-primes
  
import java.io.*;
  
class GFG {
      
    // A function to count all prime factors 
    // of a given number
    static int countPrimeFactors(int n)
    {
  
        int count = 0;
  
        // Count the number of 2s that divide n
        while (n % 2 == 0) {
  
            n = n / 2;
            count++;
        }
  
        // n must be odd at this point. So we 
        // can skip one element (Note i = i +2)
        for (int i = 3; i <= Math.sqrt(n); 
                                  i = i + 2) {
  
            // While i divides n, count i and 
            // divide n
            while (n % i == 0) {
  
                n = n / i;
                count++;
            }
        }
  
        // This condition is to handle the case 
        // whien n is a prime number greater 
        // than 2
        if (n > 2)
            count++;
  
        return (count);
    }
  
    // A function to print the first n numbers 
    // that are k-almost primes.
    static void printKAlmostPrimes(int k, int n)
    {
  
        for (int i = 1, num = 2; i <= n; num++) {
              
            // Print this number if it is k-prime
            if (countPrimeFactors(num) == k) {
                  
                System.out.print(num + " ");
  
                // Increment count of k-primes
                // printed so far
                i++;
            }
        }
  
        return;
    }
  
    /* Driver program to test above function */
    public static void main(String[] args)
    {
        int n = 10, k = 2;
        System.out.println("First " + n + " "
             + k + "-almost prime numbers : ");
  
        printKAlmostPrimes(k, n);
    }
}
  
// This code is contributed by vt_m.

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Python3

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# Python3 Program to print first 
# n numbers that are k-primes
import math
  
# A function to count all prime
# factors of a given number
def countPrimeFactors(n):
    count = 0;
  
    # Count the number of 
    # 2s that divide n
    while(n % 2 == 0):
        n = n / 2;
        count += 1;
  
    # n must be odd at this point.
    # So we can skip one
    # element (Note i = i +2)
    i = 3;
    while(i <= math.sqrt(n)):
          
        # While i divides n, 
        # count i and divide n
        while (n % i == 0):
            n = n / i;
            count += 1;
        i = i + 2;
  
    # This condition is to handle
    # the case whien n is a
    # prime number greater than 2
    if (n > 2):
        count += 1;
  
    return(count);
  
# A function to print the 
# first n numbers that are
# k-almost primes.
def printKAlmostPrimes(k, n):
    i = 1;
    num = 2
    while(i <= n):
          
        # Print this number if
        # it is k-prime
        if (countPrimeFactors(num) == k):
            print(num, end = "");
            print(" ", end = "");
  
            # Increment count of 
            # k-primes printed
            # so far
            i += 1;
        num += 1;
    return;
  
# Driver Code
n = 10
k = 2;
print("First n k-almost prime numbers:");
printKAlmostPrimes(k, n);
  
# This code is contributed by mits

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C#

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// C# Program to print first n 
// numbers that are k-primes 
using System;
  
class GFG 
{
      
    // A function to count all prime  
    // factors of a given number
    static int countPrimeFactors(int n)
    {
        int count = 0;
  
        // Count the number of 2s that divide n
        while (n % 2 == 0) 
        {
            n = n / 2;
            count++;
        }
  
        // n must be odd at this point. So we 
        // can skip one element (Note i = i +2)
        for (int i = 3; i <= Math.Sqrt(n); 
                                i = i + 2) 
        {
  
            // While i divides n, count i and 
            // divide n
            while (n % i == 0) 
            {
                n = n / i;
                count++;
            }
        }
  
        // This condition is to handle 
        // the case when n is a prime
        // number greater than 2 
        if (n > 2)
            count++;
  
        return (count);
    }
  
    // A function to print the first n
    // numbers that are k-almost primes.
    static void printKAlmostPrimes(int k, int n)
    {
  
        for (int i = 1, num = 2; i <= n; num++) 
        {
              
            // Print this number if it is k-prime
            if (countPrimeFactors(num) == k) 
            {
                   Console.Write(num + " ");
  
                // Increment count of k-primes
                // printed so far
                i++;
            }
        }
  
        return;
    }
  
    // Driver code
    public static void Main()
    {
        int n = 10, k = 2;
        Console.WriteLine("First " + n + " "
            + k + "-almost prime numbers : ");
  
        printKAlmostPrimes(k, n);
    }
}
  
// This code is contributed by Nitin Mittal.

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PHP

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<?php
// PHP Program to print first 
// n numbers that are k-primes
  
// A function to count all prime
// factors of a given number
function countPrimeFactors($n)
{
    $count = 0;
  
    // Count the number of 
    // 2s that divide n
    while($n % 2 == 0)
    {
        $n = $n / 2;
        $count++;
    }
  
    // n must be odd at this point.
    // So we can skip one
    // element (Note i = i +2)
    for ($i = 3; $i <= sqrt($n); $i = $i + 2)
    {
          
        // While i divides n, 
        // count i and divide n
        while ($n % $i == 0)
        {
            $n = $n/$i;
            $count++;
        }
    }
  
    // This condition is to handle
    // the case whien n is a
    // prime number greater than 2
    if ($n > 2)
        $count++;
  
    return($count);
}
  
// A function to print the 
// first n numbers that are
// k-almost primes.
function printKAlmostPrimes($k, $n)
{
    for ($i = 1, $num = 2; $i <= $n; $num++)
    {
          
        // Print this number if
        // it is k-prime
        if (countPrimeFactors($num) == $k)
        {
            echo($num);
            echo(" ");
  
            // Increment count of 
            // k-primes printed
            // so far
            $i++;
        }
    }
    return;
}
  
    // Driver Code
    $n = 10; 
    $k = 2;
    echo "First $n $k-almost prime numbers:\n";
    printKAlmostPrimes($k, $n);
  
// This code is contributed by nitin mittal. 
?>

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Output :

First 10 2-almost prime numbers : 
4 6 9 10 14 15 21 22 25 26 

References:
https://en.wikipedia.org/wiki/Almost_prime

This article is contributed by Rachit Belwariar. If you like GeeksforGeeks and would like to contribute, you can also write an article and mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

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Improved By : nitin mittal, Mithun Kumar